Our result considers that the controlucis supported on Γc,which is an open subset of the inflow part Γin(see (1.5)) of the boundary. This is in fact natural to control the inflow boundary of the channel. At the same time we remark that our analysis applies if one wants to control the outflow boundary Γout or the lateral boundary Γ0of the channel Ω. In what follows we briefly discuss these cases.
(i) Controlling the outflow boundary.In this case the control zone Γc is an open subset of Γout. After the change of unknowns (1.11), one can imitate the linearization procedure (as done while transforming (1.12) into (1.13)). In this linearized system the transport equation modeling the density (1.13)1–(1.13)3
will remain unchanged but the boundary conditions on the velocity equations (1.13)4–(1.13)8 should be
replaced byy = 0 on (Γ0∪Γin)×(0,∞) and y=
Nc
P
j=1
wj(t)gj(x) on Γout×(0,∞). Still the proof of the boundary stabilizability of the Oseen equations can be carried in a similar way as done in Section 2 and in the same spirit of Corollary 2.10, one can prove that if the initial condition y0 and the non-homogeneous term f are suitably small then the inflow and the outflow boundaries of the perturbed vector field (vs+e−βty) coincide with that of vs. Since the transport equation (1.13)1–(1.13)3 remains unchanged in this case, the analysis done in Section3applies without any change. The fixed point argument done in Section4 to prove the stabilization of the coupled system (1.2) also applies without change.
(ii) Controlling the lateral boundary.In this case the control zone Γcis an open subset of Γ0. In particular we assume that Γc ⊂Γb (where Γb = (0, d)× {0} ⊂Γ0). Now the inflow and outflow boundaries of the velocity vector (e−βty+vs) cannot be characterized by using the notations Γin and Γout (as defined in (1.10)), since Γc can contain an inflow part and an outflow part and one can not prove a result similar to Corollary2.10. More precisely here we can use the following notations for timet >0,
Γ∗in,y(t) = Γin∪ {x∈Γc|(vs(x) +e−βty(x, t)·n(x))<0} ⊂Γin∪Γb,
Γh= (0, d)× {1}. (5.1)
In a similar way as we have obtained (1.13) from (1.2), one gets the following system
One can use arguments similar to the ones in Section2in order to stabilizeysolving (5.2)4–(5.2)8. The functions gj can be constructed with compact support in Γb (imitating the construction (2.11)), and we can recover the C∞regularity of the boundary control andV2,1(Q∞) regularity ofy. Hence the flow corresponding to the vector field (e−βty+vs) is well defined in classical sense, consequently one can adapt the arguments used in Section3 to prove that σ,the solution of (5.2)1–(5.2)3belongs toL∞(Q∞) and vanishes after some finite time provided the initial conditionσ0 is supported away from the lateral boundaries andy is small enough. The use of a fixed point argument to prove the stabilizability of the solution of (1.2) is again a straightforward adaptation of the arguments used in Section4.
Acknowledgements. The author wishes to thank the ANR project ANR-15-CE40-0010 IFSMACS as well as the Indo-French Centre for Applied Mathematics (IFCAM) for the funding provided during this work. The author deeply thank the anonymous referee for valuable comments which helped a lot in improving the article.
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